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793

[Journal of Political Economy, 2002, vol. 110, no. 4]� 2002 by The University of Chicago. All rights reserved. 0022-3808/2002/11004-0003$10.00

Asset Pricing with Heterogeneous Consumers

and Limited Participation: Empirical Evidence

Alon BravDuke University

George M. ConstantinidesUniversity of Chicago and National Bureau of Economic Research

Christopher C. GeczyUniversity of Pennsylvania

We present evidence that the equity premium and the premium ofvalue stocks over growth stocks are consistent in the 1982–96 periodwith a stochastic discount factor calculated as the weighted averageof individual households’ marginal rate of substitution with low andeconomically plausible values of the relative risk aversion coefficient.Since these premia are not explained with an SDF calculated as theper capita marginal rate of substitution with a low value of the RRAcoefficient, the evidence supports the hypothesis of incomplete con-sumption insurance. We also present evidence that an SDF calculatedas the per capita marginal rate of substitution is better able to explainthe equity premium and does so with a lower value of the RRA co-efficient, as the definition of asset holders is tightened to recognizethe limited participation of households in the capital market.

We thank John Cochrane, Jonathan Parker, Geert Rouwenhorst, Chris Telmer, AnnetteVissing-Jørgensen, Wolf Weber, Simon Wheatley, two anonymous referees, and participantsat the American Finance Association conference, Asia-Pacific video seminar series, AtlantaFed, Centre for Interuniversity Research and Analysis on Organizations conference inMontreal, the European Financial Management Association conference, London Schoolof Economics, NBER Asset Pricing conference, Princeton University, and Yale Universityfor helpful comments.

794 journal of political economy

I. Introduction and Summary

In a representative-consumer exchange economy, one set of implicationsof the equilibrium are the Euler equations of per capita consumption.In tests of the conditional Euler equations of per capita consumption,Hansen and Singleton (1982), Ferson and Constantinides (1991), Han-sen and Jagannathan (1991), and others reject the model.

A related set of equilibrium implications that take into account boththe Euler equations of per capita consumption and the market-clearingconditions are the predictions of a calibrated economy on the uncon-ditional mean and standard deviation of the market return and the risk-free rate. Mehra and Prescott (1985) demonstrate that the equilibriumof a reasonably parameterized representative-consumer exchange econ-omy is able to furnish a mean annual premium of equity return overthe risk-free rate of, at most, 0.35 percent. This contrasts with the his-torical premium of 6 percent in U.S. data. Furthermore, as stressed inWeil (1989), the equilibrium annual risk-free rate of interest is consis-tently too high, about 4 percent, as opposed to the observed 1 percentin U.S. data.

Several generalizations of essential features of the model have beenproposed to mitigate its poor performance. They include alternativeassumptions on preferences (e.g., Abel 1990; Constantinides 1990; Ep-stein and Zin 1991; Ferson and Constantinides 1991; Benartzi and Tha-ler 1995; Daniel and Marshall 1997; Campbell and Cochrane 1999, 2000;Boldrin, Christiano, and Fisher 2001), modified probability distributionsto admit rare but disastrous marketwide events (Mehra and Prescott1988; Rietz 1988), incomplete markets (e.g., Bewley 1982; Mehra andPrescott 1985; Mankiw 1986; Telmer 1993; Lucas 1994; Constantinidesand Duffie 1996; Heaton and Lucas 1997, 2000; Storesletten, Telmer,and Yaron 1999; Krebs 2000), market imperfections (e.g., Aiyagari andGertler 1991; Danthine, Donaldson, and Mehra 1992; Brav and Geczy1995; He and Modest 1995; Bansal and Coleman 1996; Heaton andLucas 1996; Luttmer 1996; Basak and Cuoco 1998; Alvarez and Jermann2000; Constantinides, Donaldson, and Mehra 2002), and the survivalbias of the U.S. capital markets.1 Cochrane and Hansen (1992), Ko-cherlakota (1996), and Cochrane (1997) provide excellent surveys ofthis literature.

Full consumption insurance implies that heterogeneous consumersare able to equalize, state by state, their marginal rate of substitution.Therefore, the equilibrium in a heterogeneous-consumer, full-infor-mation economy is isomorphic in its pricing implications to the equi-

1 However, Jorion and Goetzmann (1999, table 6) find that the average real capital gainrate of a U.S. equities index exceeds the average rate of a global equities index thatincludes markets that both have and have not survived by merely 1 percent per year.

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librium in a representative-consumer, full-information economy if con-sumers have von Neumann–Morgenstern preferences (see Wilson 1968;Constantinides 1982). The strong assumption of full consumption in-surance is indirectly built in asset pricing models in finance and neo-classical macroeconomic models through the assumption of the exis-tence of a representative consumer.

Bewley (1982), Mehra and Prescott (1985), and Mankiw (1986) sug-gest the potential of enriching the asset pricing implications of therepresentative-consumer paradigm by relaxing the assumption of com-plete consumption insurance.2 With the exception of Constantinidesand Duffie (1996), the extant research suggests that the potential en-richment is largely illusory.3 Constantinides and Duffie find that incom-plete consumption insurance enriches substantially the implications ofthe representative-consumer model. Their main result is a propositiondemonstrating, by construction, the existence of household income pro-cesses, consistent with a given aggregate income process such that equi-librium security and bond price processes match the given security andbond price processes. Since the proposition demonstrates the existenceof equilibrium in frictionless markets, it implies that the Euler equationsof household (but not necessarily of per capita) consumption must hold.

The first goal of our paper is to examine the asset pricing implicationsof the relaxation of the assumption of complete consumption insurance.The basis of our empirical investigation is the set of Euler equations ofhousehold consumption, as opposed to the Euler equations of per capitaconsumption.4 The set of Euler equations of household consumptionimply that any household’s marginal rate of substitution and anyweighted sum thereof are a valid stochastic discount factor (SDF). Sinceindividual consumption data are reported with substantial error, it is

2 There is an extensive literature on the hypothesis of complete consumption insurance;see Cochrane (1991), Mace (1991), Altonji, Hayashi, and Kotlikoff (1992), and Attanasioand Davis (1996).

3 Telmer (1993) and Lucas (1994) calibrate economies in which consumers face un-insurable income risk and borrowing or short-selling constraints. They conclude thatconsumers come close to the complete-markets rule of complete risk sharing, althoughconsumers are allowed to trade in just one security in a frictionless market. Aiyagari andGertler (1991) and Heaton and Lucas (1996, 1997) add transaction costs or borrowingcosts or both and reach a similar negative conclusion, provided that the supply of bondsis not restricted to an unrealistically low level. The primary reason why Constantinidesand Duffie (1996) find that incomplete consumption insurance substantially enriches theasset pricing implications of the representative-consumer model is that they allow theidiosyncratic income shocks to be persistent and their conditional variance to be relatedto the state variables in a particular way, in contrast to earlier work that restricts theidiosyncratic income shocks to being transient and homoscedastic.

4 Related studies include those by Jacobs (1999), who studies the Panel Study of IncomeDynamics (PSID) database on food consumption, and Cogley (2002) and Vissing-Jørgensen(2002; this issue), who study the Consumer Expenditure Survey (CEX) database on broadmeasures of consumption.


difficult to test directly the hypothesis that each household’s marginalrate of substitution is a valid SDF. Therefore, we test the hypothesis thatthe SDF given by the equally weighted sum of the households’ marginalrates of substitution is a valid SDF.

The bulk of our tests are on the premium of the value-weighted andthe equally weighted market portfolio return over the risk-free rate. Wedo not reject the hypothesis that the equally weighted sum of the house-holds’ marginal rates of substitution is a valid SDF with a relative riskaversion (RRA) coefficient between two and four. We perform severalrobustness tests that reinforce the conclusion. An RRA coefficient be-tween two and four is economically plausible.

We investigate the properties of the cross-sectional distribution of thehousehold consumption growth that drive the SDF. We find that a Taylorexpansion of the SDF that captures the skewness, in addition to themean and variance, of the cross-sectional distribution explains the equitypremium. However, a Taylor expansion of the SDF that captures themean and variance (but not the skewness) of the cross-sectional distri-bution of the household consumption growth does not fare as well. ATaylor expansion of the SDF, in terms of the logarithm of the householdconsumption growth, that captures the mean, variance, and skewnessof the cross-sectional distribution of the household consumption growthdoes not fare well either. These results underscore the importance ofthe skewness, combined with the first two moments of the cross-sectionaldistribution. They also suggest that empirical findings based on log-linearized Euler equations of individual households should be treatedwith caution.

The second goal of our paper is to reexamine the asset pricing im-plications of the limited participation of households in the capital markets.Mankiw and Zeldes (1991), Blume and Zeldes (1993), and Haliassosand Bertaut (1995) present evidence of limited participation of house-holds in the capital markets. Specifically, they observe that only a smallfraction of individuals and households hold equities either directly orindirectly. Furthermore, Mankiw and Zeldes calculate the per capitafood consumption of a subset of households, designated as asset holdersaccording to a criterion of asset holdings above some threshold. Theyfind that the implied RRA coefficient decreases as the threshold israised.5 Attanasio and Weber (1995) argue that food consumption is adubious proxy for total consumption.

5 Brav and Geczy (1995) provide the first confirmation of the Mankiw and Zeldes (1991)results by using per capita consumption of nondurables and services reconstructed fromthe CEX database. Section V of the current paper contains an updated and extendedversion of Brav and Geczy (1995) and subsumes the 1995 draft. Related results are pre-sented in the paper by Attanasio, Banks, and Tanner (2002; this issue), who study theU.K. Family Expenditure Survey database, and Vissing-Jørgensen (2002; this issue), whostudies the CEX database.

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We recognize the fact that only a subset of households is marginal inthe stock market by defining as asset holders the subset of householdsthat report total assets exceeding a certain threshold value ranging from$0 to $40,000. From the subset of households defined as asset holders,we express the SDF in terms of the per capita growth rate and testwhether this SDF explains the equity premium. We find that the modelis better able to explain the equity premium and does so with a de-creasing value of the RRA coefficient as the definition of asset holdersis tightened. The results are sensitive to empirical design.

We also report the correlation of the per capita consumption growthwith the equity premium. There is a pattern of increasing correlationas the definition of asset holders is tightened. These results are in linewith earlier results reported by Mankiw and Zeldes (1991) and Brav andGeczy (1995). In summary, we find some evidence that the SDF drivenby the per capita consumption growth can explain the equity premiumwith a relatively high value of the RRA coefficient, once we recognizethe fact that only a subset of households is marginal in the stock market.

All the tests reported so far, whether under the hypothesis of completeor incomplete consumption insurance, focus on explaining the equitypremium. Finally, we report results of tests with the unconditional Eulerequation on the excess return of high-book-to-market “value” stocks overlow-book-to-market “growth” stocks. This may be viewed as a test of theconditional Euler equation, where the attribute of book-to-market is theconditioning variable. In addition, as in the market portfolio, both partsof this spread between value and growth are typically available to in-vestors through brokerage and retirement accounts in the form of man-aged fund investments. (We do not attempt to explain the premium ofsmall- vs. large-capitalization stocks because there is no size premiumin our sample period.) We conclude that the SDF implied by a modelof incomplete consumption insurance is consistent with the value pre-mium whereas the SDF implied by a model of complete consumptioninsurance is not. The results reinforce our earlier findings on the equitypremium.

The paper is organized as follows. In Section II, we discuss the theorythat motivates the empirical investigation. The data sources, the dataselection procedure, and summary statistics are described in Section III.In Section IV, we present the empirical results on the equity premiumunder the hypothesis of incomplete consumption insurance. In SectionV, we present the empirical results on the equity premium under thehypothesis of complete consumption insurance and examine the extentto which the equity premium is better explained by taking into consid-eration the limited participation of the households in the capital mar-kets. In Section VI, we report evidence that the premium of value stocksover growth stocks is consistent with Euler equations of consumption,


under the hypothesis of incomplete consumption insurance. In SectionVII, we provide extensions and concluding remarks.

II. The Model

A. The Economy and Equilibrium

We make conventional assumptions about the markets and preferencesin order to focus on our stated dual goal of investigating the pricingimplications of the incompleteness of markets that insure against idiosyn-cratic income shocks and the limited participation of households in thecapital markets.

We consider a set of households, that participate in thei p 1, … , I,capital markets. We assume that these households trade in perfect capitalmarkets, without frictions, short-sale restrictions, or taxes. They trade aset of securities subscripted by with total return Rj,t betweenj p 1, … , J,dates and t. We assume that the households have time- and state-t � 1separable von Neumann–Morgenstern homogeneous preferences

�

�1 t 1�aE (1 � a) b(c � 1)FF , (1)� i,t 0[ ]tp0

where a, is the constant RRA coefficient; b is the constant sub-a 1 0,jective discount factor; ci,t is the dollar consumption of the ith householdat date t; and Ft is the date t information set that is common across thehouseholds.

In equilibrium, we obtain the set of Euler equations of con-I # Jsumption between dates and t:t � 1

�aE[bg R FF ] p 1, i p 1, … , I, j p 1, … , J, (2)i,t jt t�1

where is the consumption growth of the ith household.g p c /ci,t i,t i,t�1

B. Stochastic Discount Factors

A stochastic discount factor or pricing kernel, mt, is defined by the property

E[m R FF ] p 1, j p 1, … , J. (3)t j,t t�1

We note that each household’s marginal rate of substitution,is a valid SDF, and any weighted sum of the households’�ab(c /c ) ,i,t i,t�1

marginal rates of substitution is a valid SDF also. Since individual con-sumption data are reported with substantial error, it is difficult to testdirectly the hypothesis that each household’s marginal rate of substi-tution is a valid SDF.

We may be able to mitigate the observation error in reported house-

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hold consumption by testing the hypothesis that the equally weightedsum of the households’ marginal rates of substitution is a valid SDF:

I �aci,t�1m p b I . (4)�t ( )[ ]cip1 i,t�1

This SDF is still susceptible to observation error because each term inthe sum is raised to a high power if the RRA coefficient is high.

We expand equation (4) as a Taylor series up to cubic terms. Weobtain the following approximation for the SDF:

I 2gi,t1�a �1m p bg 1 � a(a � 1)I � 1�t t ( )2[ gip1 t

I 3gi,t1 �1� a(a � 1)(a � 2)I � 1 (5)� ( )6 ]gip1 t

in terms of the cross-sectional mean, variance,�1 Ig p I � g ,ip1t i,t

and skewness, of the con-�1 2 �1 3I II � [(g /g ) � 1] , I � [(g /g ) � 1] ,ip1 ip1i,t t i,t t

sumption growth rate. We may further mitigate the observation errorif the estimation of these cross-sectional moments is less susceptible toobservation error than the SDF in equation (4). In testing the hypothesisthat the SDF is given by equation (5) against the alternative hypothesisthat the SDF is given by equation (4), we also investigate whether thecross-sectional variance and skewness of the consumption growth ratecapture most of the cross-sectional variation.

If we expand equation (4) as a Taylor series up to quadratic terms,we obtain the SDF

I 2gi,t1�a �1m p bg 1 � a(a � 1)I � 1 (6)�t t ( )2[ ]gip1 t

in terms of the average consumption growth rate and the cross-sectionalvariance of the consumption growth rate. In testing the hypothesis thatthe SDF is given by equation (6), we investigate whether the cross-sectional variance of the consumption growth rate alone captures mostof the cross-sectional variation.

If we assume that the idiosyncratic income shocks are multiplicativeand independently and identically distributed (i.i.d.) lognormal, then


Constantinides and Duffie (1996) show that the SDF in equation (4)simplifies to

�aI I� cip1 i,t 1 �1m p b exp a(a � 1)I log (g )�t ( ) i,tI 2{ [� c ip1ip1 i,t�1

2I

�1� I log (g ) . (7)� i,t ] }ip1

In testing the hypothesis that the SDF is given by equation (7), weinvestigate whether multiplicative and i.i.d. lognormal idiosyncratic in-come shocks capture most of the cross-sectional variation of the con-sumption growth rate.6

If a complete set of markets exists that enables households to insureagainst idiosyncratic income shocks, then the heterogeneous house-holds are able to equalize, state by state, their marginal rates of substi-tution. Therefore, the equilibrium of a heterogeneous-household, full-information economy is isomorphic in its pricing implications to theequilibrium of a representative-household, full-information economy(see Constantinides 1982). In particular, the consumption growth rateis identical across households, and the SDF in equation (4) simplifiesto

�am p bg . (8)t t

Market completeness also implies that the SDF in equation (4) simplifiesto

�aI� cip1 i,tm p b . (9)t ( )I� cip1 i,t�1

We expect that the SDF given by equation (9) is less susceptible toobservation error than the SDF given by equation (8).

Tests of the SDFs given by either one of equations (8) and (9) againstthe SDFs given by any of equations (4)–(7) are tests of the hypothesisof complete consumption insurance against the alternative hypothesisof incomplete consumption insurance. These tests are the focus of thepaper.

6 Krebs (2000) generalizes the lognormal idiosyncratic income process of consumers byintroducing a process that assigns probability p to an event of near personal bankruptcy:a consumer’s permanent income drops close to zero with probability p. When the per-manent income is made sufficiently close to zero in the event of near bankruptcy, theprospect of near bankruptcy does affect equilibrium prices, even if p is made arbitrarilysmall. Then idiosyncratic income shocks can have an important effect on prices, eventhough we can make the covariance between the equity premium and the cross-sectionalvariance of arbitrarily small.log (g )i,t

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C. Tests of Stochastic Discount Factors

In most of our tests, we test each candidate SDF with the unconditionalEuler equation on the excess market return, (both the equallyR � RM,t F,t

weighted and value-weighted), as

E[m (R � R )] p 0. (10)t M,t F,t

Specifically, we calculate the statistic u asT

�1u p T m (R � R ) (11)� t M,t F,ttp1

and interpret it as the unexplained mean premium.7

We also test some SDFs with the unconditional Euler equation on theexcess return of high-book-to-market (value) stocks over low-book-to-market (growth) stocks, as and calculateR � R , E[m (R � R )] p 0H,t L,t t H,t L,t

the corresponding unexplained-premium statistic asT

�1u p T m (R � R ).� t H,t L,ttp1

This may be viewed as a test of the conditional Euler equation (3), wherethe attribute of book-to-market is the conditioning variable. We do nottest the SDFs with the unconditional Euler equation on the excess returnof small- versus large-capitalization stocks because there is no size pre-mium in our sample period.

D. Observation Error in the Consumption Data

Observation error in the consumption data is a major problem both inour investigation and in related ones. In testing the Euler equations ofconsumption under the assumption of complete consumption insuranceand limited capital market participation, we calculate per capita con-sumption in a quarter as the average consumption of households thatare classified as asset holders, on the basis of a certain threshold of house-hold assets holdings. The number of households in each subsample issmall, and the estimated per capita consumption is noisy.

Observation error is even more problematic when we test the Eulerequations of consumption under the assumption of incomplete con-

7 We motivate the interpretation of the statistic u as the unexplained mean premiumby writing eq. (11) as

T T T�1

�1 �1 �10 p T m R � R � T m u ≈ T m (R � R � u)� � �( )t M,t F,t t t M,t F,t[ ]tp1 tp1 tp1

since is approximately equal to which is approximately equal to�1 �1T(T � m ) E[R ],tp1 t F,t

one.


sumption insurance. The individual household’s marginal rate of sub-stitution is calculated by raising the individual household’s consumptiongrowth to a power equal to the negative of the RRA coefficient. If thereported consumption growth of even one household out of many issubstantially smaller than one, this household’s marginal rate of sub-stitution is large and may dominate the weighted average of the marginalrates of substitution.

The standard remedy of trimming the sample of household con-sumption growth rates is a double-edged sword that we apply with cau-tion. The potentially interesting events that help distinguish betweenthe pricing implications of models of complete and incomplete con-sumption insurance are the major uninsurable shocks to a household’sincome, such as job loss or divorce. If these shocks are uninsurable,they result in household consumption growth in the tails of thedistribution.

We illustrate the implications of a multiplicative and unbiased obser-vation error in the consumption level, in the context of the hypothesisof complete consumption insurance. The SDF is given by equation (9).We assume that the observed per capita consumption is where thec w ,t t

observation error, wt, has the following properties: ; ; wtw 1 0 E[w ] p 1t t

is identically distributed, but possibly serially correlated; and wt is in-dependent of all other variables in the Euler equation. The unexplainedpremium statistic, u, in equation (11) is

T �act�1 �a au p bT (R � R )w w . (12)� M,t F,t t t�1( )ctp1 t�1

Under the null hypothesis that the Euler equation holds, the meanvalue of the statistic is zero. Therefore, observation error of the partic-ular form assumed here does not bias the unexplained risk premiumstatistic.

We also test the Euler equation on the risk-free rate, RF,t, as the realreturn on a one-month, rolled-over Treasury bill rate by testing whetherthe implied subjective discount factor, b, is close to but less than one.The estimated subjective discount factor is

�1T �act�1 �a ab̂ p T w w R (13)� t t�1 F,t( )[ ]ctp1 t�1

and is biased downward by the multiplicative factor 8�a a �1{E [w w ]} ≤ 1.t t�1

8 This inequality follows from the fact that and its inverse are symmetrically�a aw wt t�1

distributed and�a a �a a �1 �a a �a a �11 p E[(w w )(w w ) ] ≤ E[w w ]E[(w w ) ]t t�1 t t�1 t t�1 t t�1

�a a 2p {E[w w ]} .t t�1

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As predicted, the estimated subjective discount factor is severely biaseddownward. The bias renders the estimates meaningless, and therefore,they are not reported in the paper.

In the case of incomplete consumption insurance, similar argumentslead to the conclusion that observation error of the particular formassumed here does not bias the unexplained risk premium statistic butbiases downward the estimated subjective discount factor. This bias ismore severe than in the case of complete consumption insurance be-cause the observed household consumption has substantially higher errorthan the observed per capita consumption.

E. Small-Sample Properties of the Statistics

The second major problem both in our investigation and in relatedones is the small size of the database, both in the time series and inthe number of households in the cross section. The database consistsof returns and household consumption data for 60 quarters. With sucha short time series, the standard error of the estimated mean equitypremium is large, and we may be unable to reject the hypothesis thatthe mean equity premium is zero. Furthermore, we may be unable todetect the incremental contribution of relaxing the assumption of com-plete consumption insurance in explaining the equity premium.

Finally, the uninsurable idiosyncratic shocks to the households’ in-come that the theory attempts to capture, such as job loss or divorce,are infrequent events relative to both the length of the time series andthe number of households in the cross section.

In the empirical section, we address these problems by calculatingthe small-sample distribution of the F-statistic by the bootstrap methodand adjusting the p-value accordingly.

III. Description of the Data

A. The Consumption Data

The source of the household-level quarterly consumption data is theConsumer Expenditure Survey, produced by the Bureau of Labor Sta-tistics (BLS).9 This series of cross sections covers the period 1980:1–2000:4. Each quarter, roughly 5,000 U.S. households are surveyed, cho-sen randomly according to stratification criteria determined by the U.S.Census.

Each household participates in the survey for five consecutive quar-ters, one training quarter and four regular ones, during which its recent

9 Among the uses of the survey is the calculation of weights on individual componentsof the market basket of goods used in creating the consumer price index (CPI).


consumption and other information are recorded. At the end of its fifthquarter, another household, chosen randomly according to stratificationcriteria determined by the U.S. Census, replaces the household. Thecycle of the households is staggered uniformly across the quarters suchthat new households replace approximately one-fifth of the participatinghouseholds each quarter.10 If a household moves away from the sampleaddress, it is dropped from the survey. The new household that movesinto this address is screened for eligibility and is included in the survey.The number of households in the database varies from quarter toquarter.

The survey attempts to account for an estimated 95 percent of allquarterly household expenditures in each consumption category froma highly disaggregated list of consumption goods and services. At theend of the fourth regular quarter, data are also collected on the dem-ographics and financial profiles of the households, including the valueof asset holdings as of the month preceding the interview. We use con-sumption data only from the regular quarters since we consider the datafrom the training quarter unreliable. In a significant number of years,the BLS did not survey rural households. Therefore, we consider onlyurban households.

The CEX survey reports are categorized in three tranches, which weterm the January, February, and March tranches. For a given year, thefirst-quarter consumption of the January tranche corresponds to con-sumption over January, February, and March; for the February tranche,first-quarter consumption corresponds to consumption over February,March, and April; for the March tranche, first-quarter consumptioncorresponds to consumption over March, April, and May; and so on forthe second-, third-, and fourth-quarter consumption. Whereas the CEXconsumption data are presented on a monthly frequency, for someconsumption categories, the numbers reported as monthly are simplyquarterly estimates divided by three.11 Thus utilizing monthly consump-tion is not an option.

Following Attanasio and Weber (1995), we discard from our samplethe consumption data for the years 1980 and 1981 because they are ofquestionable quality. Starting in interview period 1986:1, the BLSchanged its household identification numbering system without provid-

10 If we were to exclude the training quarter in classifying a household as being in thepanel, then each household would stay in the panel for four quarters and new householdswould replace one-fourth of the participating households each quarter. The constantrotation of the panel makes it impossible to test hypotheses regarding a specific house-hold’s behavior through time for more than four quarters. A longer time series of indi-vidual households’ consumption is available from the PSID database, albeit only on foodconsumption.

11 See Attanasio and Weber (1995) and Souleles (1999) for further details regardingthe database.

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ing the correspondence between the 1985:4 and 1986:1 identificationnumbers of households interviewed in both quarters. This change inthe identification system makes it impossible to match households acrossthe 1985:4–1986:1 gap and results in the loss of some observations. Thisproblem recurs between 1996:1 and 1997:1. In this instance, we opt toend our sample in 1996:1. Thus our sample covers the period 1982:1–1996:1.

B. Definition of the Consumption Variables

For each tranche, we calculate each household’s quarterly consumptionof nondurables and services by aggregating the household’s quarterly con-sumption across the consumption categories included in the definitionof nondurables and services. We employ aggregation weights that adhereto the National Income and Product Accounts (NIPA) definitions ofconsumption of nondurables and services. In addition, we deflate eachhousehold’s consumption to the 1996:1 level, using the CPI for con-sumption of nondurables and services. We obtain the CPI series fromthe BLS through Citibase.

A household’s consumption growth between quarters and t ist � 1defined as the ratio of the household’s consumption in quarters t and

The household’s consumption growth is seasonally adjusted byt � 1.using the additive adjustments obtained from the per capita consump-tion growth, as described below.

The per capita consumption of a set of households is calculated asfollows. First, the consumption of each household is normalized bydividing it by the number of family members in the household. Second,the normalized household consumptions are averaged across the set ofhouseholds. The per capita consumption growth between quarters t �

and t is defined as the ratio of the per capita consumption in quarters1t and For each tranche, the per capita consumption growth ist � 1.seasonally adjusted by using additive adjustments obtained from regres-sion on all the quarterly consumption growths.

C. Household Selection Criteria

In any given quarter, we delete from the sample households that in thatquarter report as zero their total consumption, their consumption ofnondurables and services, or their food consumption. In any given quar-ter, we also delete from the sample households with missing informationon these items.

We define a household’s beginning total assets as the sum of the house-hold’s market value of stocks, bonds, mutual funds, and other securities


at the beginning of the first regular quarter.12 We define as asset holders thehouseholds that report total assets exceeding a certain threshold.We present results for threshold values ranging from $0 to $40,000 in1996:1 dollars. The number of households that are included as assetholders in our sample varies across quarters and across thresholds.

We mitigate observation error by subjecting the households to a con-sumption growth filter. The filter consists of the following three selectioncriteria. First, we delete from the sample households with consumptionreported in fewer than three consecutive quarters. Second, we deletethe consumption growth if and Third, we1c /c c /c ! c /c 1 2.i,t i,t�1 i,t i,t�1 i,t�1 i,t2delete the consumption growth if it is greater than five. Thec /ci,t i,t�1

surviving subsample of households is substantially smaller than the orig-inal one.

In table 1, we present summary statistics on the quarterly per capitaconsumption of nondurables and services for the period 1982:1–1996:1, in 1996:1 dollars, for a variety of definitions of asset holders.Given that we drop quarters for which the consumption growth filteris undefined, there are 52 usable quarters in the 1982:1–1996:1 period.Per capita consumption is obtained from the CEX, with asset holdersdefined as the households in the database that report total assets, in1996:1-adjusted dollars, satisfying the listed criterion and satisfying theconsumption-growth filter. We present summary statistics separately foreach of the three tranches, January, February, and March.

Among all the tranches, the total number of households with anyamount of assets ranges between 533 and 825. The number of house-holds that are classified as asset holders diminishes rapidly as the thresh-old value is raised. Among all the tranches and across time, the numberof households with assets exceeding $2,000 ranges between 30 and 113,and the number of households with assets exceeding $20,000 rangesbetween 13 and 71. A high threshold in the definition of asset holderseliminates households that are inframarginal in the capital markets butdecreases the number of households in the database. We recognize thistrade-off by presenting empirical results for a wide range of thresholdvalues.

The standard deviation of the per capita consumption growth rate islarge, reflecting the fact that the number of households in each sub-sample is small. For some subsamples, the sample mean of the per capitaconsumption growth is negative but well within one standard deviationfrom zero.

12 During the fifth and last interview, the household is asked to report both the end-of-period asset holdings and the change in these asset holdings relative to a year earlier.From this we calculate the household’s asset holdings at the beginning of the first regularquarter.

asset pricing 807

TABLE 1Summary Statistics on per Capita Quarterly Consumption

Number of HouseholdsHousehold Con-sumption Level

Household Con-sumption

Growth Rate

Minimum Median Maximum MeanStandardDeviation Mean

StandardDeviation

A. Total Assets ≥$0

January 552 692 825 2,437 368 �.01 .06February 569 682 761 2,466 370 �.01 .07March 533 688 794 2,436 378 �.01 .06

B. Total Assets ≥$2,000

January 31 80 108 3,351 528 .00 .08February 30 81 104 3,426 554 �.01 .09March 39 81 113 3,469 606 .00 .09

C. Total Assets ≥$10,000

January 22 53 80 3,541 560 .00 .10February 18 54 76 3,621 583 �.02 .10March 23 56 83 3,665 631 �.01 .11

D. Total Assets ≥$20,000

January 14 40 69 3,657 605 .00 .10February 13 40 63 3,764 606 �.02 .10March 16 40 71 3,773 681 �.01 .12

Note.—We present summary statistics on the quarterly per capita consumption of nondurables and services byhouseholds for the period 1982:1–1996:1. The household’s consumption of nondurables and services is calculated byaggregating the household’s quarterly consumption across the consumption categories that constitute the definitionof nondurables and services. We employ aggregation weights that adhere to the NIPA definitions of consumption ofnondurables and services. The household consumption data are filtered using the methods described in Sec. IIIC andare deflated to the 1996:1 level, using the CPI for consumption of nondurables and services. We obtain the CPI seriesfrom the BLS through Citibase. We report sample means and standard deviations for both the level of consumptionand consumption growth for a variety of definitions of asset holders as well as summary statistics on the number ofobservations in the particular asset-holding layer. Asset holders are defined as the households in the database that reporttotal assets, in 1996-adjusted dollars. We present summary statistics separately for each of the three tranches (interviewgroups) labeled January, February, and March.

D. The Returns Data

Our measure of the nominal, monthly risk-free rate of interest is theone-month Treasury bill return. We calculate the three-month nominalreturn as the compounded buy-and-hold, three-month return. The realquarterly risk-free rate is calculated as the nominal risk-free rate, dividedby the three-month (one plus) inflation rate, based on the deflatordefined for nondurables and services.

The value-weighted nominal, monthly market return (capital gainplus dividends) is an arithmetic return. It is calculated from the pooledsample of stocks listed on the New York Stock Exchange and the Amer-ican Stock Exchange, obtained from the Center for Research in SecurityPrices of the University of Chicago. We calculate the nominal, quarterlymarket return as the compounded buy-and-hold, three-month invest-


ment. We calculate the real quarterly market return as the nominalmarket return, divided by the three-month (one plus) rate of inflation.Finally, we calculate the quarterly premium on the value-weighted port-folio as the difference between the real quarterly market return andthe real quarterly interest rate. We also report results using the equallyweighted market return.

We calculate the excess return of high-book-to-market versus low-book-to-market stocks as in Fama and French (1993). The excess returnis the difference of the return of the high-book-to-market and low-book-to-market portfolios.

IV. Empirical Results on the Equity Premium under IncompleteConsumption Insurance

A. The Main Results

We begin by testing the hypothesis that the equally weighted sum ofthe households’ marginal rates of substitution is a valid SDF. Specifically,we test the hypothesis that the SDF, given by equation (4), satisfiesequation (10) on the equally weighted and on the value-weighted marketpremia. We set the subjective discount factor equal to one and considervalues of the RRA coefficient in the range zero to nine. We calculatethe unexplained premium statistic, u, as in equation (11) over the period1982:1–1996:1 and test the hypothesis that the unexplained premiumequals zero.

In panel A of table 2, we report the unexplained premium of thevalue-weighted market portfolio for each of the three tranches separatelyand for the combined tranches. We discuss first the results for the threetranches separately. We calculate the standard error of the unexplainedpremium as the sample standard deviation of the time-series observa-tions of the quantity We report the p-value of the nullm (R � R ).t M,t F,t

hypothesis against an unspecified alternative, based on the t-u p 0statistic.

In the first row, the RRA coefficient is set equal to zero, and therefore,the SDF is identically equal to one. The unexplained premium is thesample mean of the entire market premium. For the January tranche,the premium is 2.10 percent per quarter and is statistically significantwith a p-value of 2 percent. The premium is significant for the Februaryand March tranches also. Thus there is a premium that needs to beexplained in the sample period, and this observation motivates thesearch for a suitable SDF.

In the second to tenth rows, we report the unexplained premium andthe p-value of the null hypothesis For each of the tranches, theu p 0.unexplained premium becomes statistically insignificant when the RRA

TABLE 2Unexplained Equity Premium under Incomplete Consumption Insurance

RRA

Combined Tranches January Tranche February Tranche March Tranche

AverageUnexplained

PremiumF-Statistic

p-Value

F-StatisticBootstrap

p-Value

AverageUnexplained

Premiumt-Statistic

p-Value

AverageUnexplained

Premiumt-Statistic

p-Value

AverageUnexplained

Premiumt-Statistic

p-Value

A. Value-Weighted Equity Premium

0 1.85 .04 .06 2.10 .02 2.07 .02 2.46 .011 1.95 .04 .06 2.33 .01 2.12 .02 2.56 .022 2.32 .06 .07 3.02 .01 2.42 .05 2.99 .023 1.88 .74 .69 3.85 .11 2.36 .20 1.50 .314 !�10 .52 .59 !�10 .60 !�10 .74 !�10 .845 !�10 .28 .52 !�10 .67 !�10 .91 !�10 .886 !�10 .24 .54 !�10 .64 !�10 .95 !�10 .917 !�10 .23 .53 !�10 .65 !�10 .94 !�10 .898 !�10 .24 .58 !�10 .69 !�10 .94 !�10 .899 !�10 .26 .63 !�10 .69 !�10 .94 !�10 .90

B. Equally Weighted Equity Premium

0 1.78 .21 .22 2.09 .06 1.98 .07 2.27 .041 1.83 .24 .24 2.38 .04 1.95 .08 2.28 .062 2.00 .36 .29 3.05 .04 2.04 .12 2.31 .103 �.20 .86 .81 2.53 .28 .77 .39 �4.63 .744 !�10 .31 .57 !�10 .70 !�10 .88 !�10 .955 !�10 .20 .56 !�10 .76 !�10 .96 !�10 .996 !�10 .18 .55 !�10 .83 !�10 .95 !�10 .997 !�10 .17 .59 !�10 .84 !�10 .97 !�10 1.008 !�10 .19 .57 !�10 .86 !�10 .97 !�10 1.009 !�10 .21 .59 !�10 .87 !�10 .97 !�10 1.00

Note.—We calculate the unexplained premium statistic, u, on the basis of the SDF in eq. (4) over the period 1982:1–1996:1 and test the hypothesis that the unexplained premiumequals zero. p-values of the null hypothesis against an unspecified alternative are based on either asymptotic normality or bootstrapped distribution (see Sec. IVA).u p 0


coefficient becomes three and crosses zero between the values of threeand four. The sign of the unexplained premium becomes negative foran RRA coefficient of four or higher. Therefore, we do not reject thehypothesis that the equally weighted sum of the households’ marginalrate of substitution is a valid SDF with RRA coefficient equal to three.An RRA coefficient of this order of magnitude is economically plausible.

A standard generalized method of moments (GMM) estimate of riskaversion in the exactly identified case can be inferred from table 2 andothers that report unexplained premia for various levels of risk aversion.In an exactly identified GMM risk aversion estimation, the weightingmatrix essentially plays no role. The sole determinant of the risk aversionestimate then is pricing error, the squared function of which GMMminimizes. This same estimate can be read off of our unexplained pre-mium tables with a negligible amount of eyeball interpolation. For in-stance, the value-weighted premium unexplained under complete con-sumption insurance in panel A of table 2 crosses a value of zero (forthe combined tranches) for an RRA between three and four. For theequally weighted case, the unexplained premium crosses zero betweenvalues of two and three for the RRA, although probably closer to threethan two.

Note that each household in the sample is represented in only oneof the three tranches. If a household’s consumption growth is an outlier,this outlier cannot influence the results in more than one of thetranches. The fact that the estimated unexplained premia and the p-values are very similar for the three tranches is evidence that the resultsare robust to observation error on the households’ consumption growth.

We also report the unexplained premium for the combined tranches.The unexplained premium is calculated as the weighted average of theunexplained premia of the three tranches, where the weights are de-termined from the weighted least squares quadratic form

where u3 is a vector of estimated unexplained′ �1 �1 ′(1 V 1 ) 1 u , 3 # 13 3 3 3

premia for the three tranches, V is the diagonal of the covariance3 # 3matrix, and 13 is a unit vector.13 We thus weight each mean by a3 # 1measure of its volatility. Unweighted arithmetic means produce quali-tatively similar results, as means calculated using the entire covariancematrix (generalized least squares) do. We calculate the F-statistic and

13 We note that as a result of the identification problem that hampers the ability tomatch households between 1985:4 and 1986:1, individual tranches have different time-series lengths. This difference affects the calculation of the combined average unexplainedpremia and the F-statistic and bootstrap p-values reported. These quantities are calculatedfor the combined cases using the time frame common for all tranches. This explains why,in some cases, the combined premia fall outside the range of the individual trancheestimates. An alternative would be to truncate ex ante the time series for all tranches toa common time frame. We eschew this approach, however, since it discards informationunnecessarily.

asset pricing 811

report the p-value of the null hypothesis that the combined unexplainedpremium is zero. We also calculate the small-sample distribution of theF-statistic by the bootstrap method and report the p-value. Specifically,the F-statistic is the Hotelling test of the null that the mean unex-2Tplained premia are jointly zero: ′ �1 �1[T(T � 3)/3(T � 1)](u V u ) ∼3 3

We utilize a block bootstrap with a block size of four quarters.F .3,T�3

In the first row of panel A of table 2, the sample mean of the entirepremium for the combined tranches is 1.85 percent per quarter and ismarginally significant. In the second to tenth rows, we report the un-explained premium and the p-value of the null hypothesis foru p 0increasing values of the RRA coefficient. The unexplained premiumbecomes statistically insignificant when the RRA coefficient becomesthree, and the sign of the unexplained premium becomes negative foran RRA coefficient of four or higher, consistent with the results for theindividual tranches.

In panel B of table 2, we report the unexplained premium of theequally weighted market portfolio for each of the three tranches sep-arately and for the combined tranches. The sample mean of the entirepremium for the combined tranches is 1.78 percent per quarter, butwith a p-value of about 20 percent, it is not significant. For the individualtranches, the premium is significant at the 10 percent level. For thethree tranches separately and for the combined tranches, the unex-plained premium reverses sign when the RRA coefficient is either threeor four. Overall, the pattern of the unexplained premia of the equallyweighted market portfolio is consistent with the earlier results on thevalue-weighted market portfolio. This suggests that the results are robustto outliers in the portfolio returns and to the composition of the marketportfolio.

B. Robustness of the Results

We explore further the robustness of the empirical results presented intable 2. We expand the SDF as a Taylor series up to cubic terms, as inequation (5), and test the hypothesis that the expanded SDF satisfiesequation (10) on the value-weighted and on the equally weighted marketpremia. The SDF is expressed in terms of the cross-sectional mean,variance, and skewness of the household consumption growth rate. Themotivation for this procedure is that the estimation of the cross-sectionalmoments may be less susceptible to outliers than the estimation of theSDF in equation (4): the estimates of the cross-sectional moments areindependent of the RRA coefficient, whereas the SDF in equation (4)is very sensitive to outliers in the household consumption growth whenthe RRA coefficient is large.

The results are reported in table 3 and are similar to the results

TABLE 3Arithmetic Expansion of the SDF under Incomplete Consumption Insurance

RRA

Combined Tranches January Tranche February Tranche March Tranche

AverageUnexplained

PremiumF-Statistic

p-Value


p-Value

AverageUnexplained

Premiumt-Statisticp-Value

AverageUnexplained


AverageUnexplained



0 1.85 .04 .05 2.10 .02 2.07 .02 2.46 .011 1.76 .03 .05 2.07 .01 1.91 .02 2.32 .012 1.51 .02 .03 1.80 .01 1.56 .04 1.98 .013 .90 .05 .03 1.05 .01 .86 .07 1.21 .014 �.22 .96 .97 �.40 .77 �.30 .68 �.15 .575 �1.85 .44 .43 �2.74 .96 �1.99 .89 �2.26 .796 �4.12 .32 .26 �6.16 .98 �4.27 .90 �5.23 .837 �7.12 .27 .27 !�10 .98 �7.16 .90 �9.22 .838 !�10 .26 .21 !�10 .99 !�10 .89 !�10 .849 !�10 .25 .24 !�10 .99 !�10 .88 !�10 .87


0 1.78 .21 .20 2.09 .06 1.98 .07 2.27 .041 1.68 .21 .21 2.11 .04 1.80 .08 2.12 .052 1.46 .15 .17 1.84 .04 1.50 .08 1.84 .023 .97 .12 .07 1.02 .06 .98 .08 1.32 .024 �.15 .87 .90 �.64 .81 .17 .44 .48 .405 �1.59 .65 .57 �3.38 .94 �.96 .67 �.73 .616 �3.61 .56 .53 �7.46 .97 �2.44 .71 �2.31 .677 �6.29 .51 .45 !�10 .98 �4.29 .73 �4.29 .698 !�10 .48 .40 !�10 .98 �6.53 .71 �6.67 .689 !�10 .46 .39 !�10 .98 �9.22 .68 �9.47 .68

Note.—We calculate the unexplained premium statistic, u, on the basis of the SDF in eq. (5) over the period 1982:1–1996:1 and test the hypothesis that the unexplainedpremium equals zero. The SDF employed here is expressed in terms of the cross-sectional mean, variance, and skewness of the household consumption growth rate. Seealso the note to table 2.

asset pricing 813

presented in table 2. For each of the three tranches separately and forthe combined tranches, the unexplained, value-weighted equity pre-mium is negative for an RRA coefficient of four or higher. We obtainsimilar results for the equally weighted equity premium. For the Januarytranche and for the combined tranches, the sign change occurs for anRRA coefficient equal to four; for the January and February tranches,the sign change occurs for an RRA coefficient equal to five.

For a high-RRA coefficient, the unexplained premium in table 3 isnegative but not as negative as in table 2. This is consistent with theexplanation that the estimation of the cross-sectional moments is lesssusceptible to outliers than the estimation of the SDF in equation (4).

The finding that the SDF as in equation (5) explains the equity pre-mium with only a slightly higher RRA coefficient than the SDF as inequation (4) suggests that the cross-sectional variance and skewness ofthe household consumption growth rate capture most of the cross-sec-tional variation of the households’ consumption growth rates.

C. The Role of the Cross-Sectional Skewness of the Consumption GrowthRate

We investigate the role of the cross-sectional skewness of the householdconsumption growth rate in explaining the equity premium by ex-panding the SDF as a Taylor series up to quadratic terms, as in equation(6), and testing the hypothesis that the expanded SDF satisfies equation(10) on the value-weighted and on the equally weighted market premia.The SDF is expressed in terms of the cross-sectional mean and variance,but not skewness, of the household consumption growth rate. Thus theSDF differs from the SDF of equation (5) only in that the skewness ofthe cross-sectional consumption growth rate is suppressed.

The results are reported in columns 1–3 of table 4, panels A and B.The unexplained value-weighted equity premium not only remains pos-itive for all values of the RRA coefficient between zero and nine butalso increases as the RRA coefficient increases. However, the unex-plained premium is insignificant at the 5 percent level for an RRAcoefficient of two or higher, with p-values ranging to 10 percent for anRRA coefficient equal to nine. The unexplained equally weighted equitypremium also remains positive and increasing for all values of the RRAcoefficient between zero and nine but is not statistically significant.These results underscore the importance of the skewness of the house-hold consumption growth rate, combined with the first two momentsof the cross-sectional distribution, in explaining the equity premium.

We explore further the importance of the skewness by testing a variantof the expansion of the SDF, given by equation (5). We define G pi,t

as the logarithmic consumption growth of the ithlog (c ) � log (c )i,t i,t�1

TABLE 4Additional Tests under Incomplete Consumption Insurance

RRA

SDF: Eq. (6) SDF: Eq. (7) SDF: Eq. (8) SDF: Eq. (9)

AverageUnexplained

Premium(1)

F-Statisticp-Value

(2)


p-Value(3)

AverageUnexplained

Premium(4)

F-Statisticp-Value

(5)


p-Value(6)

AverageUnexplained

Premium(7)

F-Statisticp-Value

(8)


p-Value(9)

AverageUnexplained

Premium(10)

F-Statisticp-Value

(11)


p-Value(12)


0 1.85 .04 .05 1.85 .04 .05 1.85 .04 .05 1.85 .05 .0191 1.97 .04 .04 1.94 .04 .06 1.73 .04 .04 1.85 .04 .0192 2.32 .05 .07 2.30 .05 .06 1.62 .04 .06 1.87 .05 .0193 2.84 .06 .09 3.09 .07 .08 1.53 .04 .07 1.90 .07 .0194 3.53 .07 .09 4.70 .10 .10 1.46 .04 .04 1.93 .04 .0195 4.35 .07 .08 8.08 .16 .15 1.39 .04 .04 1.98 .05 .0206 5.30 .08 .10 110 .22 .22 1.33 .04 .06 2.04 .06 .0207 6.37 .08 .10 110 .30 .29 1.28 .04 .06 2.11 .06 .0218 7.54 .08 .09 110 .36 .39 1.23 .04 .07 2.19 .06 .0229 8.81 .08 .10 110 .39 .46 1.20 .04 .06 2.28 .06 .023


0 1.78 .21 .22 1.78 .21 .23 1.78 .21 .20 1.78 .21 .201 1.87 .24 .23 1.83 .24 .25 1.65 .22 .22 1.77 .22 .242 2.15 .27 .25 2.11 .29 .23 1.54 .22 .22 1.78 .22 .223 2.60 .30 .27 2.69 .35 .29 1.45 .22 .21 1.80 .22 .214 3.19 .31 .30 3.81 .44 .33 1.38 .21 .19 1.83 .22 .225 3.90 .32 .28 5.94 .51 .40 1.31 .21 .20 1.87 .21 .206 4.73 .32 .28 110 .55 .43 1.26 .20 .17 1.92 .21 .217 5.67 .31 .24 110 .54 .47 1.21 .20 .18 1.98 .20 .218 6.71 .30 .28 110 .49 .48 1.17 .19 .17 2.05 .19 .179 7.85 .29 .25 110 .42 .44 1.13 .18 .15 2.12 .19 .17

Note.—We calculate the unexplained premium statistic, u, on basis of the four different SDFs over the period 1982:1–1996:1 and test the hypothesis that the unexplained premium equalszero. The first SDF, given in eq. (6), is expressed in terms of the cross-sectional mean and variance of the household consumption growth rate. The second SDF is based on the assumptionthat idiosyncratic income shocks that are multiplicative and i.i.d. lognormal drive the cross-sectional variation of the households’ consumption growth rates (see eq. [7]). The third and fourthSDFs are based on the complete-markets assumption and are given by eqq. (8) and (9), respectively. See also the note to table 2.

asset pricing 815

household. We expand equation (4) as a Taylor series up to cubic terms.We obtain the following approximation for the SDF,

I I1 1�aG 2 �1 2 3 �1 3tm p be 1 � a I (G � G ) � a I (G � G ) , (14)� �t i,t t i,t t2 6[ ]

ip1 ip1

in terms of the cross-sectional mean, variance,�1 IG p I � G ,ip1t i,t

and skewness, of the logarithmic�1 2 �1 2I II � (G � G ) , I � (G � G ) ,ip1 ip1i,t t i,t t

consumption growth rate. In empirical results that we do not displayhere, we find that the SDF given by equation (14) fails to explain theequity premium. These results contrast with the results reported in Sec-tion IVB that the SDF given by equation (5) explains the equity pre-mium. We surmise that, in expanding the SDF in terms of the logarith-mic consumption growth rate, we suppress the effect of outliers and,in particular, suppress the effect of the skewness on the SDF.14 Theseresults underscore further the importance of the skewness on the SDF.

D. The Role of the Lognormality Assumption of the HouseholdConsumption Growth Rate

We investigate whether multiplicative and i.i.d. lognormal idiosyncraticincome shocks capture the cross-sectional variation of the consumptiongrowth rate. Under this assumption, the SDF is given by equation (7).We test the hypothesis that this SDF satisfies equation (10) on the value-weighted and on the equally weighted market premia.

The results are reported in columns 4–6 of table 4, panels A and B.The unexplained value-weighted equity premium remains positive forall values of the RRA coefficient between zero and nine and increasesas the RRA coefficient increases. However, the unexplained premiumis marginally insignificant at the 5 percent level for an RRA coefficientof one or higher. The unexplained equally weighted equity premiumalso remains positive for all values of the RRA coefficient between zeroand nine but is not statistically significant. Contrasted with the resultson the SDF given by equations (4) and (5), these results again under-score the importance of the skewness of the household consumptiongrowth rate, combined with the first two moments of the cross-sectionaldistribution, in explaining the equity premium.

14 We explore this assertion by calculating the simple correlation between the cross-sectional mean, variance, and skewness based on gi, the simple consumption growth ofthe ith household, and Gi, the logarithm of the consumption growth of the ith household.For both the cross-sectional mean and variance, we find correlations that, in general,exceed 90 percent between the two possible ways of computing these sample moments.However, the time-series sample estimates of skewness have a much lower correlation (31percent, 49 percent, and 18 percent for the January, February, and March tranches,respectively).


V. Empirical Results on the Equity Premium under CompleteConsumption Insurance

A. Tests of Complete Consumption Insurance

If a complete set of markets exists that enables households to insureagainst idiosyncratic income shocks, then the heterogeneous house-holds are able to equalize, state by state, their marginal rates of substi-tution. Then the SDF may be expressed in terms of the cross-sectionalmean, but not skewness and variance, of the household consumptiongrowth rate, as in equation (8). It may also be expressed in terms ofthe per capita growth rate, as in equation (9). We test the hypothesisof complete consumption insurance by testing whether these two SDFssatisfy equation (10) on the value-weighted and on the equally weightedmarket premia.

The results are reported in columns 7–12 of table 4, panels A and B.The unexplained value-weighted equity premium remains positive andstatistically significant for all values of the RRA coefficient between zeroand nine. This particular SDF fails to explain the equity premium.

B. Tests of Complete Consumption Insurance with Limited Stock MarketParticipation

We recognize the fact that only a subset of households is marginal inthe stock market by defining as asset holders the subset of householdsthat report total assets exceeding a certain threshold value ranging from$0 to $40,000. From the subset of households defined as asset holders,we express the SDF in terms of the per capita growth rate, as in equation(9), and repeat the tests of complete consumption insurance.

For threshold values $0, $2,000, $10,000, $20,000, $30,000, and$40,000, the unexplained quarterly value-weighted equity premium re-mains positive and statistically significant for all values of the RRA co-efficient between zero and 20. These results, not reported here, suggestthat the assumption of complete consumption insurance that suppressesthe cross-sectional variation of the households’ consumption growth ratefails to explain the quarterly equity premium even after we take intoaccount the limited stock market participation.

We repeat the tests but now switch the holding period from three tosix months. The results are reported in table 5. For threshold value $0,the unexplained six-month value-weighted equity premium is 5.18 per-cent and is statistically significant at all levels of the RRA coefficient.For threshold value $2,000, the unexplained premium drops to 3.37percent and becomes statistically insignificant when the RRA coefficientequals 10. For threshold value $10,000, the unexplained premium dropsto 3.07 percent and becomes statistically insignificant when the RRA

TABLE 5Unexplained Equity Premium under Complete Consumption Insurance and

Limited Participation

RRA u StatisticF-Statistic

p-ValueF-Statistic

Bootstrap p-Value

A. Threshold Value ≥$0

0 5.18 .00 .001 5.21 .00 .005 5.29 .00 .0710 5.00 .01 .3015 4.86 .02 .3120 4.83 .04 .37

B. Threshold Value ≥$2,000

0 5.18 .00 .001 4.91 .00 .005 4.08 .02 .0210 3.37 .20 .3415 2.84 .57 .6420 2.50 .79 .83

C. Threshold Value ≥$10,000

0 5.18 .00 .001 4.89 .00 .005 3.97 .03 .0310 3.07 .31 .4715 2.09 .81 .8320 .51 .94 .93

D. Threshold Value ≥$20,000

0 5.18 .05 .031 4.82 .00 .005 3.65 .06 .0510 2.02 .77 .8015 �.74 .98 .9720 �5.80 .80 .85

E. Threshold Value ≥$30,000

0 5.18 .00 .001 4.82 .00 .005 4.00 .09 .0710 3.48 .50 .5715 2.31 .70 .7820 �2.43 .65 .74

F. Threshold Value ≥$40,000

0 5.18 .00 .001 4.71 .00 .005 3.21 .18 .1510 1.49 .82 .8315 �.38 .92 .9020 �3.87 .88 .87

Note.—We calculate the unexplained premium statistic, u, on the basis of the SDF in eq. (9) over the period 1982:1–1996:1 and test the hypothesis that the unexplained premium equals zero. We report the unexplained premiumbased on the value-weighted market portfolio and a semiannual compounding frequency for the combined threetranches. We report the values of the statistic u for the combined portfolio of tranches. See also the note to table 2.


coefficient equals 10. For threshold values $20,000 and $40,000, theunexplained premium flips sign for RRA coefficients between 10 and15; for threshold value $30,000, the unexplained premium flips sign forRRA coefficients between 15 and 20. We interpret these results as evi-dence of complete consumption insurance when limited market par-ticipation is taken into account, if one considers these values of theRRA coefficient economically plausible. We repeat the tests but shift thestart of the six-month holding period by one quarter. In results notreported here, we find that we are unable to explain the equity premiumfor any threshold value between $0 and $40,000 and for any RRA co-efficient between zero and 20. We conclude that, while we uncover primafacie evidence in favor of complete consumption insurance with limitedparticipation, the results are sensitive to the empirical design.

In table 6, we report the correlation of the per capita consumptiongrowth with the equity premium on the value-weighted and the equallyweighted market indices at the six-month frequency. For each thresholdvalue, we report the correlation for each of the January, February, andMarch tranches separately since there is no obvious way to combinethese correlation estimates across tranches. Whereas the correlation es-timates differ widely across the tranches, there is a pattern of increasingcorrelation as the definition of asset holders is tightened. We stress thatwe make no claim of statistical significance in this pattern since we donot report p-values in table 6. We make statements backed with statisticalsignificance in the context of the unexplained premium statistic. Nev-ertheless, these results are in line with earlier results reported by Mankiwand Zeldes (1991) and Brav and Geczy (1995) and recent results byAttanasio et al. (2002; this issue) and Vissing-Jørgensen (2002; this issue).

In summary, we find some evidence that the SDF driven by the percapita consumption growth can explain the equity premium with a rel-atively high value of the RRA coefficient, once we recognize the factthat only a subset of households is marginal in the stock market.

VI. Explaining the Premium of Value Stocks over Growth Stocks

All the tests reported so far pertain to the equity premium. In thissection, we report results of tests with the unconditional Euler equationon the excess return of high-book-to-market value stocks over low-book-to-market growth stocks, as and calcu-R � R , E[m (R � R )] p 0H,t L,t t H,t L,t

late the corresponding unexplained-premium statistic as u pThis may be viewed as a test of the conditional�1 TT � m (R � R ).tp1 t H,t L,t

Euler equation (3), where the attribute of book-to-market is the con-ditioning variable. We do not attempt to explain the premium of small-versus large-capitalization stocks because there is no size premium inour sample period.

TABLE 6Calibration Results under Complete Consumption Insurance and Limited

Participation

JanuaryTranche

FebruaryTranche

MarchTranche

A. Total Assets ≥$0

Correlation with value-weighted market .07 .22 �.15Correlation with equally weighted market .07 .19 �.21RRA:

Value-weighted market 269 98 �35Equally weighted market 184 73 �18

B. Total Assets ≥$2,000

Correlation with value-weighted market .22 .54 .32Correlation with equally weighted market .09 .52 .25RRA:

Value-weighted market 28 14 28Equally weighted market 53 9 26

C. Total Assets ≥$10,000



D. Total Assets ≥$20,000



E. Total Assets ≥$30,000



F. Total Assets ≥$40,000



Note.—We calculate the correlation and implied risk aversion parameter, as in Mankiw and Zeldes (1991), of theper capita consumption growth with the equity premium on the equally weighted and value-weighted market indicesat the six-month frequency over the period 1982:1–1996:1.


The results are reported in table 7 for the combined tranches. In thefirst row, the RRA coefficient is set equal to zero, and therefore, theSDF is identically equal to one. The unexplained premium is the samplemean of the entire value premium. The unexplained value premium is1.19 percent and is significant at the 10 percent level.

In the second to tenth rows, we report the unexplained premium andthe p-value of the null hypothesis under different assumptionsu p 0on the SDF. In columns 1–3, we report the unexplained value premiumwhen the SDF is given by equation (4), under the assumption of in-complete consumption insurance. The sign of the unexplained pre-mium is reversed when the RRA coefficient lies between three and four.In columns 4–6, we report the unexplained value premium when theSDF is expressed in terms of the cross-sectional mean, variance, andskewness of the household consumption growth, as given by equation(5). The sign of the unexplained premium is reversed when the RRAcoefficient lies between four and five.

By contrast, in columns 7–9, we report the unexplained value pre-mium when the SDF is expressed in terms of the per capita consumptiongrowth, as implied by the hypothesis of complete consumption insur-ance and as given by equation (9). The unexplained premium is positive,although marginally insignificant, and increases as the value of the RRAcoefficient increases. We conclude that the SDF implied by a model ofincomplete consumption insurance is consistent with the value pre-mium, whereas the SDF implied by a model of complete consumptioninsurance is not. The results reinforce our earlier findings on the equitypremium.

VII. Extensions and Concluding Remarks

In this paper, we presented evidence that the equity premium and thepremium of value stocks over growth stocks are explained with an SDFcalculated as the weighted average of the individual households’ mar-ginal rate of substitution with low and economically plausible values ofthe RRA coefficient. Since the premia above are not explained with anSDF calculated as the per capita marginal rate of substitution with lowvalues of the RRA coefficient, the evidence supports the hypothesis ofincomplete consumption insurance. The results are robust across thevalue-weighted equity premium, the equally weighted equity premium,and the value-over-growth premium. The results are robust across thethree distinct cohorts of households that we refer to as the January,February, and March tranches.

Now that the cross-sectional variance and, particularly, the cross-sec-tional skewness of the households’ consumption growth rates have beenidentified as important components of the SDF, it is of interest to in-

TABLE 7Unexplained Premium of Value Stocks over Growth Stocks

RRA

SDF: Eq. (4) SDF: Eq. (5) SDF: Eq. (9)

AverageUnexplained

Premium(1)

F-Statisticp-Value

(2)


p-Value(3)

AverageUnexplained

Premium(4)

F-Statisticp-Value

(5)


p-Value(6)

AverageUnexplained

Premium(7)

F-Statisticp-Value

(8)


p-Value(9)

0 1.19 .05 .10 1.19 .05 .10 1.19 .05 .101 1.30 .05 .11 1.19 .05 .10 1.22 .05 .122 1.67 .05 .11 1.13 .04 .06 1.26 .05 .093 2.93 .21 .24 .91 .04 .07 1.29 .05 .114 �.15 .73 .79 .50 .57 .54 1.33 .06 .125 !�10 .68 .79 �.22 .98 .97 1.38 .06 .126 !�10 .63 .77 �1.26 .78 .71 1.43 .07 .117 !�10 .59 .75 �2.68 .63 .58 1.48 .08 .128 !�10 .56 .75 �4.52 .55 .52 1.54 .08 .129 !�10 .54 .73 �6.82 .51 .46 1.61 .09 .15

Note.—We calculate the unexplained premium statistic, u, on the basis of the SDFs given in eqq. (4), (5), and (9) over the period 1982:1–1996:1 and test the hypothesisthat the unexplained premium equals zero. We report the unexplained premium on the excess return of high-book-to-market “value” stocks over low-book-to-market “growth”stocks for the combined tranches. The factor time series were kindly provided by Gene Fama and Ken French.


vestigate the comovement of these moments with macroeconomicvariables.

We also tested the implications of the ubiquitous complete consump-tion insurance model, the “representative-consumer” model, when thelimited participation of households in the capital markets is taken intoaccount. Consistent with earlier results, we reject the model when noprovision is made for the limited participation. We also presented someevidence that an SDF calculated as the per capita marginal rate of sub-stitution is better able to explain the equity premium and does so witha lower value of the RRA coefficient as the definition of asset holdersis tightened to recognize the limited participation of households in thecapital market. Our evidence that the representative-consumer modelcan account for the equity premium when limited participation is takeninto account is sensitive to the experimental design. It is of interest toexplore further the robustness of this evidence.

References

Abel, Andrew B. “Asset Prices under Habit Formation and Catching Up withthe Joneses.” A.E.R. Papers and Proc. 80 (May 1990): 38–42.

Aiyagari, S. Rao, and Gertler, Mark. “Asset Returns with Transactions Costs andUninsured Individual Risk.” J. Monetary Econ. 27 (June 1991): 311–31.

Altonji, Joseph G.; Hayashi, Fumio; and Kotlikoff, Laurence J. “Is the ExtendedFamily Altruistically Linked? Direct Tests Using Micro Data.” A.E.R. 82 (De-cember 1992): 1177–98.

Alvarez, Fernando, and Jermann, Urban J. “Efficiency, Equilibrium, and AssetPricing with Risk of Default.” Econometrica 68 (July 2000): 775–97.

Attanasio, Orazio P.; Banks, James; and Tanner, Sarah. “Asset Holding and Con-sumption Volatility.” J.P.E. 110 (August 2002): 771–92.

Attanasio, Orazio P., and Davis, Steven J. “Relative Wage Movements and theDistribution of Consumption.” J.P.E. 104 (December 1996): 1227–62.

Attanasio, Orazio P., and Weber, Guglielmo. “Is Consumption Growth Consistentwith Intertemporal Optimization? Evidence from the Consumer ExpenditureSurvey.” J.P.E. 103 (December 1995): 1121–57.

Bansal, Ravi, and Coleman, Wilbur John, II. “A Monetary Explanation of theEquity Premium, Term Premium, and Risk-Free Rate Puzzles.” J.P.E. 104 (De-cember 1996): 1135–71.

Basak, Suleyman, and Cuoco, Domenico. “An Equilibrium Model with RestrictedStock Market Participation.” Rev. Financial Studies 11 (Summer 1998): 309–41.

Benartzi, Shlomo, and Thaler, Richard H. “Myopic Loss Aversion and the EquityPremium Puzzle.” Q.J.E. 110 (February 1995): 73–92.

Bewley, Truman F. “Thoughts on Tests of the Intertemporal Asset Pricing Model.”Working paper. Evanston, Ill.: Northwestern Univ., 1982.

Blume, Marshall E., and Zeldes, Stephen P. “The Structure of Stock Ownershipin the U.S.” Working paper. Philadelphia: Univ. Pennsylvania, 1993.

Boldrin, Michele; Christiano, Lawrence J.; and Fisher, Jonas D. M. “Habit Per-sistence, Asset Returns, and the Business Cycle.” A.E.R. 91 (March 2001):149–66.

Brav, Alon, and Geczy, Christopher C. “An Empirical Resurrection of the Simple

asset pricing 823

Consumption CAPM with Power Utility.” Working paper. Chicago: Univ. Chi-cago, 1995.

Campbell, John Y., and Cochrane, John H. “By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior.” J.P.E. 107 (April1999): 205–51.

———. “Explaining the Poor Performance of Consumption-Based Asset PricingModels.” J. Finance 55 (December 2000): 2863–78.

Cochrane, John H. “A Simple Test of Consumption Insurance.” J.P.E. 99 (Oc-tober 1991): 957–76.

———. “Where Is the Market Going? Uncertain Facts and Novel Theories.” Fed.Reserve Bank Chicago Econ. Perspectives 21 (November–December 1997): 3–37.

Cochrane, John H., and Hansen, Lars Peter. “Asset Pricing Explorations forMacroeconomics.” In NBER Macroeconomics Annual 1992, edited by Olivier J.Blanchard and Stanley Fischer. Cambridge, Mass.: MIT Press, 1992.

Cogley, Timothy. “Idiosyncratic Risk and the Equity Premium: Evidence fromthe Consumer Expenditure Survey.” J. Monetary Econ. 49 (March 2002):309–34.

Constantinides, George M. “Intertemporal Asset Pricing with HeterogeneousConsumers and without Demand Aggregation.” J. Bus. 55 (April 1982):253–67.

———. “Habit Formation: A Resolution of the Equity Premium Puzzle.” J.P.E.98 (June 1990): 519–43.

Constantinides, George M.; Donaldson, John B.; and Mehra, Rajnish. “JuniorCan’t Borrow: A New Perspective on the Equity Premium Puzzle.” Q.J.E. 117(February 2002): 269–96.

Constantinides, George M., and Duffie, Darrell. “Asset Pricing with Heteroge-neous Consumers.” J.P.E. 104 (April 1996): 219–40.

Daniel, Kent, and Marshall, David. “Equity-Premium and Risk-Free-Rate Puzzlesat Long Horizons.” Macroeconomic Dynamics l, no. 2 (1997): 452–84.

Danthine, Jean-Pierre; Donaldson, John B.; and Mehra, Rajnish. “The EquityPremium and the Allocation of Income Risk.” J. Econ. Dynamics and Control 16(July–October 1992): 509–32.

Epstein, Larry G., and Zin, Stanley E. “Substitution, Risk Aversion, and theTemporal Behavior of Consumption and Asset Returns: An Empirical Anal-ysis.” J.P.E. 99 (April 1991): 263–86.

Fama, Eugene F., and French, Kenneth R. “Common Risk Factors in the Returnson Stocks and Bonds.” J. Financial Econ. 33 (February 1993): 3–56.

Ferson, Wayne E., and Constantinides, George M. “Habit Persistence and Du-rability in Aggregate Consumption: Empirical Tests.” J. Financial Econ. 29 (Oc-tober 1991): 199–240.

Haliassos, Michael, and Bertaut, Carol C. “Why Do So Few Hold Stocks?” Econ.J. 105 (September 1995): 1110–29.

Hansen, Lars Peter, and Jagannathan, Ravi. “Implications of Security MarketData for Models of Dynamic Economies.” J.P.E. 99 (April 1991): 225–62.

Hansen, Lars Peter, and Singleton, Kenneth J. “Generalized Instrumental Var-iables Estimation of Nonlinear Rational Expectations Models.” Econometrica 50(September 1982): 1269–86.

He, Hua, and Modest, David M. “Market Frictions and Consumption-Based AssetPricing.” J.P.E. 103 (February 1995): 94–117.

Heaton, John, and Lucas, Deborah J. “Evaluating the Effects of IncompleteMarkets on Risk Sharing and Asset Pricing.” J.P.E. 104 (June 1996): 443–87.


———. “Market Frictions, Savings Behavior, and Portfolio Choice.” Macroeco-nomic Dynamics 1, no. 1 (1997): 76–101.

———. “Portfolio Choice and Asset Prices: The Importance of EntrepreneurialRisk.” J. Finance 55 (June 2000): 1163–98.

Jacobs, Kris. “Incomplete Markets and Security Prices: Do Asset-Pricing PuzzlesResult from Aggregation Problems?” J. Finance 54 (February 1999): 123–63.

Jorion, Philippe, and Goetzmann, William N. “Global Stock Markets in the Twen-tieth Century.” J. Finance 54 (June 1999): 953–80.

———. “The Equity Premium: It’s Still a Puzzle.” J. Econ. Literature 34 (March1996): 42–71.

Krebs, Tom. “Consumption-Based Asset Pricing with Incomplete Markets.” Man-uscript. Providence, R.I.: Brown Univ., 2000.

Lucas, Deborah J. “Asset Pricing with Undiversifiable Risk and Short Sales Con-straints: Deepening the Equity Premium Puzzle.” J. Monetary Econ. 34 (De-cember 1994): 325–41.

Luttmer, Erzo G. J. “Asset Pricing in Economies with Frictions.” Econometrica 64(November 1996): 1439–67.

Mace, Barbara J. “Full Insurance in the Presence of Aggregate Uncertainty.”J.P.E. 99 (October 1991): 928–56.

Mankiw, N. Gregory. “The Equity Premium and the Concentration of AggregateShocks.” J. Financial Econ. 17 (September 1986): 211–19.

Mankiw, N. Gregory, and Zeldes, Stephen P. “The Consumption of Stockholdersand Nonstockholders.” J. Financial Econ. 29 (March 1991): 97–112.

Mehra, Rajnish, and Prescott, Edward C. “The Equity Premium: A Puzzle.” J.Monetary Econ. 15 (March 1985): 145–61.

———. “The Equity Premium: A Solution?” J. Monetary Econ. 22 (July 1988):133–36.

Rietz, Thomas A. “The Equity Risk Premium: A Solution.” J. Monetary Econ. 22(July 1988): 117–31.

Souleles, Nicholas S. “The Response of Household Consumption to Income TaxRefunds.” A.E.R. 89 (September 1999): 947–58.

Storesletten, Kjetil; Telmer, Chris I.; and Yaron, Amir. “Asset Pricing with Idio-syncratic Risk and Overlapping Generations.” Working paper. Pittsburgh: Car-negie Mellon Univ., 1999.

Telmer, Chris I. “Asset-Pricing Puzzles and Incomplete Markets.” J. Finance 48(December 1993): 1803–32.

Vissing-Jørgensen, Annette. “Limited Asset Market Participation and the Elas-ticity of Intertemporal Substitution.” J.P.E. 110 (August 2002): 825–53.

Weil, Philippe. “The Equity Premium Puzzle and the Risk-Free Rate Puzzle.” J.Monetary Econ. 24 (November 1989): 401–21.

Wilson, Robert B. “The Theory of Syndicates. ” Econometrica 36 (January 1968):119–32.

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